This is just my take on the matter, based on my understanding of what these terms mean:

Prayer implies a duality—One is addressing something or someone outside one’s self. There is the one who’s praying, and the one being prayed to. As a communication, it has content, a message.

Meditation, at least as I practice it, is about transcending duality by bringing attention to bear. At its best, there is no one meditating, and no object of meditation. The idea of separation between the two is let go of. It is often devoid of content; just pure experience. This is useful only in that it accesses the world of seamless wholeness, which is a very illuminating experience. It has no other “use” per se, but that doesn’t mean it’s worthless.

Critical thinking is analytical, which by its very nature breaks experience down conceptually into component parts. This is useful in dealing with the world of things and individuals, but can’t offer any insight into the world of wholeness.

Prayer comforts (and to some, deceives), meditation is a bunch of different things and it depends on who you ask, and deep critical thinking isn’t much different than critical thinking, you are just thinking more about something to come to a conclusion.

This is a great question. Personally I think they are all very similar. They require focus and clearing ones mind of a lot of distractions and static. I think meditation and prayer especially are basically the same, while analytical thought is working on solving something, but still takes focus. I know people who say many things changed for them when they began to pray, but I think that has to do with the focus it brought to their lives.

Prayer is the ‘sender’ half of communication with God. The etymology of the word classifies it as a sort of requesting.

Meditation, in the Biblical sense (i.e. “meditate on Scripture”) is rather like a cross between prayer and standard literary critical thinking. It is one of the more common ways of getting a response to the above prayer.

Critical thinking may come into play at any point, but is most obvious in apologetics.